ErNi₃A1₉, M_r = 586.22, trigonal, new type, hR78, R32 - f²edc⁴ (No. 155), a = 7.2716 (5), c=27.346 (3) A, V = 1252.2 (2)A³, Z = 6, D_x = 4.664 Mg m ^-3, A(Mo Ka) = 0.71073/k, u = 17.654 mm^-1, F(000) = 1614, T = 293 K, wR = 0.021 for 785 contributing unique reflections. The structure is built up of three kinds of monoatomic layer perpendicular to c: Al-atom layers with triangular mesh, Ni-atom layers with triangular mesh and layers containing two rare-earth-metal atoms for one Al-atom triangle. GdNi₃A1₉ [a = 7.3006 (9), c = 27.478 (5)/A, wR = 0.020 for 791 reflections] is isotypic. YNi₃AI₉ [a = 7.2894 (7), c = 27.430 (5) A, wR =0.035 for 635 reflections] and DyNi₃AI₉ [a = 7.2723 (9), c = 27.344 (6)A, wR = 0.027 for 682 reflections] crystallize in related structures with a partly disordered arrangement of Al-atom triangles and rare-earth-metal atoms. Similar monoatomic layers to those in ErNi₃AI₉ build up the orthorhombic structure of Y₂Co₃AI₉ (Y₂Co₃Ga₉ type). The stoichiometry 1:3:9 is related to that of RzT3A19 by replacing every second mixed layer of composition R₂AI₃ by an Al-atom layer (A1₃): 6RT₃A1₉ = 6R₂T₃AI₉ - 3R₂AI₃ + 3A1₃.